Let AB be taken as unity, and let it be required to multiply BD by BC. I have only to join the points A and C, and draw DE parallel to CA; then BE is the product of BD and BC. Show that the length of BE is the product of the lengths BD and BC.
I have no idea
To show that the length of BE is equal to the product of the lengths BD and BC, we can use the concept of similar triangles and the properties of parallel lines.
1. First, let's draw a diagram to visualize the given information. Start with points A, B, and C as mentioned, where AB is taken as unity (or as 1 unit).
A ________ C
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D| /E
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B
2. Join the points A and C to form line AC.
3. Draw a parallel line to AC passing through point B. This will create line segment DE.
4. Now, we have two triangles, ΔBDE and ΔABC, with corresponding sides:
BD and BC (common side),
DE and AC (parallel sides).
5. By the property of similar triangles, we know that corresponding sides of similar triangles are proportional.
6. Therefore, we have the proportion: BD/DE = BC/AC.
7. Rearranging the equation, we get: DE = (BD * AC) / BC.
8. Since we took AB as unity (or as 1 unit), AC is simply BC.
9. Substituting BC for AC in the equation above, we get: DE = (BD * BC) / BC.
10. Simplifying further, we have: DE = BD.
11. From the diagram, we can see that segment BE is equal to DE.
12. Therefore, BE = BD.
13. This implies that the length of BE is equal to the product of the lengths BD and BC.
Hence, we have shown that the length of BE is indeed equal to the product of the lengths BD and BC in the given construction.